Position uncertainties of AGATA pulse-shape analysis estimated via the boostrapping method
M. Siciliano, J. Ljungvall, A. Goasduff, A. Lopez-Martens, M. Zieli?ska
aa r X i v : . [ nu c l - e x ] J a n Position uncertainties of AGATA pulse-shape analysis estimated via thebootstrapping method
M. Siciliano, ∗ J. Ljungvall, A. Goasduff,
3, 4
A. Lopez-Martens, M. Zieli´nska, and AGATA and OASIS collaborations Irfu/DPhN, CEA, Universit´e Paris-Saclay, Gif-sur-Yvette, France IJC Lab, CNRS/IN2P3, Universit´e Paris-Saclay, Orsay, France Dipartimento di Fisica e Astronomia, Universit´a di Padova, Padova, Italy INFN, Sezione di Padova, Padova, Italy
The unprecedented capabilities of state-of-the-art segmented germanium-detector arrays, such asAGATA and GRETA, derive from the possibility of performing pulse-shape analysis. The com-parison of the net- and transient-charge signals with databases via grid-search methods allows theidentification of the γ -ray interaction points within the segment volume. Their precise determina-tion is crucial for the subsequent reconstruction of the γ -ray paths within the array via trackingalgorithms, and hence the performance of the spectrometer. In this paper the position uncertaintyof the deduced interaction point is investigated using the bootstrapping technique applied to Coradioactive-source data. General features of the extracted position uncertainty are discussed as wellas its dependence on various quantities, e.g. the deposited energy, the number of firing segmentsand the segment geometry.
PACS numbers: 02.70.Rr, 06.20.Dk, 07.05.Kf, 07.85.Nc, 29.40.Gx
I. INTRODUCTION
For more than 50 years, High-Purity Germanium(HPGe) detectors have been at the heart of powerful γ -ray spectrometers aiming at unraveling the complexstructure of the atomic nucleus. Recent developments indetector technology, data acquisition and processing ledto a refinement of the detection techniques, making itpossible to deduce position information from the signalshapes of semiconductor detectors. Electrical segmenta-tion of a HPGe detector not only allows us to improvethe angular granularity and the position sensitivity at thelevel of the physical segmentation, but it permits to de-duce precise information on the γ -ray interaction pointsfrom a comparison of signals induced in the neighbor-ing segments. In order to perform such a pulse-shapeanalysis, segment signals need to be acquired with highresolution, high bandwidth and high sampling frequencyover a meaningful time period. Thanks to these techno-logical developments, new generation HPGe arrays, e.g.AGATA [1] and GRETA [2], are being constructed, witha capability to determine the position of charge gener-ation inside the detector volume from the pulse-shapeinformation.The Advanced GAmma Tracking Array [1, 3] collab-oration aims at the construction of a European 4 π γ -ray tracking array, which will constitute an unparalleledtool for investigation of nuclear structure of both stableand exotic species via high-resolution γ -ray spectroscopy.After the first campaign of the AGATA demonstratorin Legnaro National Laboratories (LNL) [4], the array ∗ Present address: Argonne National Laboratory, Lemont (IL),United States moved to GSI [5, 6] to take advantage of the radioactive-ion beams during the PreSPEC campaign, and then toGANIL [7] where it has reached a nearly 1 π configurationwith 42 HPGe crystals. In the near future, AGATA willreturn to LNL for an experimental campaign with stablebeams delivered by the XTU-TANDEM/ALPI/PIAVEaccelerator complex and with exotic beams produced bythe SPES facility. The excellent efficiency and energyresolution of the array, together with its outstanding po-sition resolution, in particular when coupled with γ -raytracking algorithms, contributed to the rich physics out-put of these experimental campaigns.In order to identify the position where a γ -ray has in-teracted inside an AGATA segmented detector, the pulse-shape analysis (PSA) is performed by comparing the ex-perimental signals with a basis of position-dependent pul-ses. The reference pulses are presently generated via sim-ulations using the AGATA Detector Library (ADL) [8].This procedure requires very accurate information on theproperties of each individual detector, such as the geom-etry, the space-charge distribution, crystal-axis orienta-tion, cross-talk properties and the response function ofthe electronics [9–13] and yields a set of simulated signalsassociated with every point of a 2-mm step grid. More-over, several groups have been working on the characteri-zation of AGATA detectors with scanning tables [14–21],in order to produce a realistic basis of pulses, independentfrom simulation assumptions and free of a potential biasintroduced by boundary conditions. Following the proce-dures outlined in Refs. [22–24], a new experimental basiswas recently developed using radioactive-source data [25].Such pulses can be extracted ad-hoc for each experimentduring the calibration and they take into account the realexperimental conditions, including alterations of the sig-nals due to the electronic response or specific featuresof individual detectors. However, a slightly worse per-formance observed for such an experimentally deducedbasis [25] suggests that they are less accurate than theADL bases.During the PSA process, the comparison of measured( m ) and simulated ( s ) signals is done by minimizing thefollowing figure of merit (FoM): F oM = X j,t i | A j,m ( t i ) − A j,s ( t i ) | p , (1)where A j,m ( t i ) and A j,s ( t i ) are the amplitudes at the t i -th sample of the signal from segment j . In the sum onlythe 56-samples traces for the segment measuring the netcharge, for the neighboring ones (typically 4-5 segments)and the core are considered. As shown in Figure 1, suchshort traces include all features of the signal that are nec-essary for the analysis of the pulse shape: the baseline (8samples), the rising part of the signal ( ≈
20 samples) andits saturation. In the FoM of Equation 1, the exponent p depends on the adopted metric. Studies of the optimiza-tion of the Doppler correction led to the adopted value p = 0 . F oM = X j w j X t i | A j,m ( t i ) − A j,s ( t i ) | p j (2)where the weighting coefficient w j and the metric expo-nent p j have different values for the transient signals andfor those of the hit segment and the core. This alterna-tive definition enables taking into account the differentposition sensitivity of the two types of signals. In fact,the signals of the hit segment and of the core changerapidly with the radial position of the interaction point,while the transient signals provide a more precise infor-mation on the φ and Z coordinates. The proposed FoMgives better performance of the PSA procedure, but itis worth noting that the best results were obtained as-suming crystal and electronics parameters (e.g. chargecarrier mobilities, electronics transfer function) differentfrom those measured and commonly used.Since PSA does not make use of the χ minimiza-tion to identify the γ -ray interaction points, standardstatistical procedures cannot be used to infer the posi-tion uncertainty. This information is crucial for futuredevelopments of the tracking algorithms, as it would al-low assigning proper weights to the determined interac-tion points when reconstructing the γ -ray path inside aHPGe array. In addition, the energy and segment de-pendence of the position uncertainty may be used to op-timize the PSA algorithms, e.g. by preventing them fromlooking for the interaction point with a precision betterthan what can be reached, in order to reduce the com-putational load of the procedure. The PSA is currentlythe main factor limiting the counting rate that can beaccepted by data acquisition and online analysis, and an improved throughput is therefore of great interest. Theposition resolution of AGATA detectors has been studiedin multiple works via different techniques [28–31], withthe resulting full width at half maximum (FWHM) ofabout 5 mm for an energy deposition of 1332.5 keV. A re-cent study based on the Doppler-correction capabilities ofthe AGATA-VAMOS++ experimental setup [32] yieldedan average value about 16% larger than the previous es-timations resulting from dedicated experiments. On theother hand, the position resolution estimated from ex-perimental data does not correspond to the position res-olution needed to optimize the performance of the γ -raytracking. In the Orsay Forward Tracking algorithm [33],indeed, the position resolution enters as a parameter thatwas optimized on data simulated assuming a position res-olution of 5 mm FWHM. However, this parameter has tobe increased by a factor of 3 [25] when tracking experi-mental data, suggesting a worse position resolution. Thiseffect may be related to a non-Gaussian distribution ofthe position uncertainties, which would lead to a largervalue of the effective position resolution required by γ -ray tracking algorithms. However, the actual shape ofthe uncertainty distribution is not known.In this context, this article presents an application ofthe bootstrapping technique to pulse-shape source data,providing an uncertainty estimate for γ -ray interactionpositions in AGATA detectors for the commonly usedPSA grid search. After discussing the differences betweenthe Full and
Adaptive grid search, the dependence ofthe position uncertainty on various variables, such as thedeposited energy, the number of firing segments withinthe same HPGe crystal and the segment geometry, isstudied for the latter.
II. METHODS
Bootstrapping is a statistical technique based on ran-dom sampling with replacement, used to estimate sta-tistical properties of a distribution, such as its stan-dard deviation, when the shape of the distribution is un-known [34, 35]. This method can be used to constructhypothesis tests, in particular when parametric inferenceis impossible or impractical due to the complicated for-mulas it would require. The basic idea of bootstrappingis that inference about a distribution from sample datacan be modeled by resampling the data and perform-ing inference about the sample from the resampled data.Since the distribution is generated from the data itselfand the investigated hypothesis (in this case: the entirePSA procedure), this statistical approach is completelyindependent from external assumptions or boundary con-ditions.In addition to its simplicity, the possibility to eval-uate the position uncertainty along the three axes in-dependently represents a great advantage of the boot-strapping technique with respect to the former studies.For instance, the Doppler-correction approach is unable C5 A m p li t ud e [ a . u . ] -20020 0 16 32 48 R es i du a l s [ a . u . ] Sample [10 ns] C6
0 16 32 48 D5
0 16 32 48 D6
0 16 32 48 E5
0 16 32 48 E6
0 16 32 48
FIG. 1: (Color online) (top) Comparison between a measured pulse (blue points and line) and the corresponding PSA-basissignal resulting from the grid search (red line). The traces are reported for the firing segment (D6) and those of the neighboringsegments that are considered in the PSA procedure. (bottom) Residuals given by the sample-by-sample difference between themeasured pulse and the basis signal. to discriminate between different positions characterizedby the same γ -ray emission angle, resulting in a possi-ble underestimation of the position resolution. Moreover,a great advantage of the bootstrapping technique is itswide applicability. In fact, contrary to other approachesthat require specific experiments or complementary de-tectors, the discussed method can be applied to any kindof data. Since the measured traces are usually registeredfor AGATA in-beam experiments and radioactive-sourceruns, the position uncertainties can be evaluated for eachdata set independently by applying the bootstrappingtechnique during the PSA procedure.The results presented in this manuscript are basedon the data collected with a Co radioactive source,placed at the center of the AGATA array. Althoughsimulations suggest that neutron damage has very mi-nor influence on the PSA performance [36], two HPGecrystals have initially been considered: crystal C013 (ina good condition) and crystal B004 (with severe neu-tron damage). A standard indicator of neutron damageis the full width at tenth of maximum (FWTM). TheFWTM/FWHM ratio for an ideal Gaussian is 1.8, whilethe average FWTM/FWHM values measured at 1332.5keV for C013 and B004 were around 1.9 and 4.2, re-spectively. The data were collected during the AGATAexperimental campaign at GANIL (for crystal C013 inJune 2017, and for crystal B004 in February 2020) andthe local-level processing preceding the PSA followed thestandard procedures, described in detail in Ref. [32]. Thepresent study showed no noticeable difference betweenthe PSA performance for C013 and B004, and thus onlythe results for C013 are presented in the following. As shown in Figure 1, the grid search identifies thePSA basis signal that minimizes the FoM of Equation 1,providing the ~r i ≡ ( X, Y, Z ) i position vector ( X , Y , and Z refer to the intrinsic reference system of each AGATAcrystal, as shown in Figure 2). From the comparisonbetween the measured pulse and the energy-normalizedbasis, the residuals can be calculated. For each origi-nal event, a new population of pulses was generated byadding a random value of the residuals to the simulatedpulse on a sample-by-sample basis. In order to avoidbias caused by a possible presence of systematic devia-tions between the measured traces and the basis, eachsegment is treated independently by considering only itsown 56-samples set of residuals, which was randomly cho-sen without repetition. Moreover, in order to ensurethe asymptotic convergence of the results, the resampledpopulation has to be sufficiently large, which means thatthe bootstrapping procedure needs to be repeated mul-tiple times. In the present study, 1000 new pulses weregenerated for every original pulse. This means that, de-spite its simplicity, the bootstrap procedure can be verydemanding in terms of time and memory, in particularwhen the generated traces are written on disk. There-fore, an additional constraint was required in order tolimit the number of iterations. To account for the en-ergy dependence of position uncertainties discussed inRef. [30], 8 different energy ranges have been defined be-tween 8 keV and 2048 keV (see Figure 4) and for eachof them an upper threshold of 10 bootstrapping eventswas imposed. This level of statistics was found sufficientto estimate the PSA uncertainty and its properties asa function of the deposited γ -ray energy. Finally, the FIG. 2: (Color online) Sketch of an AGATA detector, showingthe (top) front and (bottom) lateral view. Blue arrows repre-sent the internal frame of reference of the PSA coordinates.Nominal dimensions of the segments (black dotted lines, ar-rows and labels) are expressed in millimeters and comparedwith the actual geometry (red dashed lines). Figure adaptedfrom Ref. [1]. “bootstrap” traces were processed by the PSA, provid-ing a new set of ~r i,j ≡ ( X, Y, Z ) i,j position coordinates,where the index j runs over all “bootstrap” traces. Thesecoordinates were compared with those obtained from theoriginal traces. III. RESULTS
The position fluctuations ~ ∆ i,j ≡ ~r i − ~r i,j are obtainedfor each coordinate ( X, Y, Z ) set and the uncertainties aredefined as the standard deviation of their distributions.Information on the position uncertainty is deduced in-dependently for the three coordinates by studying thedistribution of the fluctuations along each axis, i.e. ∆ X ,∆ Y and ∆ Z (for clarity, the index i, j are dropped when the coordinate is explicitly given).The ∆ X , ∆ Y and ∆ Z fluctuations were investigated asa function of the segment geometry and position. Takinginto account the shape of the detector and the symme-try of the electric field (see Figure 2), one would expectthe fluctuation distributions to be very similar for the X and Y directions. Such a symmetry was indeed ob-served and the variations of the ∆ X , ∆ Y and ∆ Z fluctu-ations were only due to the orientation of the segmentswith respect to the frame of reference. A radial fluctua-tion ∆ XY = p ∆ X + ∆ Y was, therefore, introduced. Asshown in the left panel of Figure 3, the standard deviation σ XY remains rather constant and the small variations re-flect the tapering of the HPGe crystals. In contrast, theuncertainty along the Z axis ( σ Z ) increases with the in-teraction depth (see right panel of Figure 3). This trend,which is consistent with the results of Ref. [30, Fig.14],may have two origins. On one hand, the segment heightincreases from the front to the back of the detector [1,Fig.4], so the range of the ∆ Z fluctuations increases aswell, being less restrictive for the bottom segments. Onthe other hand, since the radioactive γ -ray source wasplaced at the center of the reaction chamber, the seg-ments at the back of the crystal were reached predomi-nantly by high-energy γ rays, which resulted in a higherCompton-scattering probability and consequently in alarger number of firing segments (i.e. those measuringa net charge). When the σ Z values are normalized to thenominal height of the segments (see Figure 2), the result-ing ratios are rather constant except for the front andback segments, where an increase is observed. For theformer, the approximately 30% excess can be explainedby the segment geometry, since the height of the frontsegments is not uniform (see Figure 8). For the backsegments, instead, the increase mostly affects the uncer-tainties evaluated for the events where the number offiring segments is ≥
3, which is in agreement with thehypothesis on the γ -ray source position.Figure 3 shows also the dependence of position uncer-tainties on the number of firing segments. When onlyone segment per crystal was measuring a net charge, thepositions obtained via the bootstrapping procedure areunequivocally equal to those determined from the originaltrace. This can be related to the fact that the signals con-stituting the database of reference were simulated assum-ing that only one segment fired. Thus, for these events,the FoM manifests a minimum so deep that the smallresiduals are negligible with respect to the difference be-tween two neighboring grid points. This hypothesis wastested by introducing a random jitter of few samples (upto ±
50 ns) in the generated traces, which again led to novariations in the position fluctuation distributions. An-other argument in its favor is the observed increase of theuncertainty with the number of firing segments. Indeed,a comparison of one-hit basis signals with experimentaltraces, which result from an overlap of different signals,yields residuals that are much larger than the statisticalfluctuations or than the electronic noise, and comparable
Adaptive σ XY [ mm ] Segment
1 2 3 4 5 6
Full
Segment fire=5fire=4fire=3fire=2
Adaptive σ Z [ mm ] Segment
1 2 3 4 5 6
Full
Segment fire=5fire=4fire=3fire=2
FIG. 3: (Color online) Position uncertainties as a function of the number of firing segments within the same crystal, for the(left) radial and (right) Z directions and for the different segment geometries. The results are reported for the Adaptive and
Full grid search of AGATA PSA. to the signals themselves. Excluding the case of only onefiring segment, the relation between the number of firingsegments ( n ) and the position resolution ( W p ≡ FWHM ≈ . · σ i ) was studied with the empirical formula W p = f + f √ n ( n ≥ . (3)The f and f parameters were obtained for the Adaptive -grid results of Figure 3 and their values are summarizedin Table I.
TABLE I: Position resolution as a function of the numberof firing segments. For each segment geometry and for thewhole detector, the values of the standard deviation shown inFigure 3 are fitted with an empirical relation given by Equa-tion 3.Segment
XY Zf [mm] f [mm] f [mm] f [mm]1 3.09 (57) 4.22 (24) 1.27 (26) 1.41 (9)2 3.74 (49) 4.07 (21) 0.68 (24) 2.03 (9)3 4.59 (80) 4.03 (35) 1.55 (26) 2.12 (9)4 2.85 (47) 5.04 (19) 1.51 (49) 2.52 (21)5 4.52 (89) 4.14 (38) 1.60 (61) 2.24 (26)6 2.85 (49) 4.99 (21) 2.17 (85) 2.83 (35)Total 3.60 (42) 4.43 (19) 0.45 (26) 2.57 (12) Figure 3 presents also a comparison of the results ob-tained with the
Adaptive [37] and the
Full grid searchmodes: in the former, the PSA is first performed on asparse grid of positions with a second step on the finergrid around the best FoM found on the sparse grid, whilein the latter the search involves all points in the finer grid,giving the absolute minimum of the FoM. The uncertain-ties estimated with the
Full grid search are systemati-cally lower than those obtained in the
Adaptive mode,but the observed difference is below 10%. This small im-provement may be meaningless considering that, due to a larger number of investigated points of the grid, boththe PSA and bootstrapping procedures are very time-consuming in the
Full mode, which takes around 10 timeslonger than the
Adaptive approach. Since such differencebetween the results of the two approaches has been ob-served for all investigated properties of the PSA-positionuncertainty, in the following only the results of the
Adap-tive grid-search mode are presented. ( ∆ X + ∆ Y + ∆ Z ) / [ mm ] Energy [keV]
BootstrappingSoderstrom -3 -2 -1 C oun t s / mm / ke V FIG. 4: (Color online) Position fluctuations as a function ofthe deposited energy. The curves represent the standard de-viation as a function of energy, given by the bootstrappingtechnique (black dashed line) and the work of S¨odestr¨om [30](red dotted line). Gamma-ray energies are limited to a rangethat is relevant for the present analysis, since the data werecollected with a Co radioactive source. For each γ -ray en-ergy the maximum intensity was normalized to 1 in order tosimplify the comparison. As introduced before, several works have been alreadydevoted to the study of the position resolution of theAGATA detector [28–32]. In particular, the study ofS¨oder-str¨om and collaborators [30], performed by com-paring Mon-te-Carlo simulations with an in-beam exper-
FIG. 5: (Color online) Position uncertainties for the (left) radial and (right) Z direction, for different energy ranges and segmentgeometries. The uncertainties are given by the standard deviation, assuming a symmetric distribution and using the Adaptive grid search. iment, discusses the dependence of the PSA uncertaintyon energy. Developing a new method based on the en-ergy resolution of Doppler-corrected γ -ray spectra, theauthors proposed an empirical relation between the po-sition resolution ( W p ) and the deposited energy ( E p ): W p ( E p ) = w + w s keVE p , (4)where the parameters w = 2 . w = 6 . γ -ray transi-tions in the 200-4000 keV energy range.The dependence of the PSA uncertainty on the de-posited energy has also been investigated via the boot-strapping technique. The evolution of the total positionfluctuations ∆ ≡ p ∆ X + ∆ Y + ∆ Z was studied as afunction of the deposited energy. As shown in Figure 4,the standard deviation of the distribution rapidly de-creases with increasing energy for E γ <
200 keV and be-comes asymptotically constant for higher energies. Sincein the current study the position resolution displays thesame behavior as the one given by the empirical relationof Ref. [30], the FWHM values obtained via the boot-strapping technique were fitted with Equation 4, yield-ing w = 4 . w = 5 . . γ -ray transitions emitted in-flight by the nucleus, and the results are systematicallyaffected by the Doppler-correction and γ -ray tracking al-gorithms. In fact, for the position resolution deduced from the Doppler correction, the sensitivity to the inter-action depth in the crystal is limited, as various depthscorrespond to the same γ -ray emission angle. In contrast,the bootstrapping approach can provide an almost con-tinuous dependence of the position uncertainty on energyfor each of the three directions, since it is only based onthe deposited energy, independently of its origin.Figure 5 presents the evolution of the position uncer-tainty for different energy ranges for the radial directionand along the Z axis. The different energy ranges wereselected as a compromise between the level of statisticsneeded to accurately investigate the deposited-energy de-pendence, and the required number of computational it-erations. Also in this case the position resolution for thehorizontal plane is rather constant for each energy range.Along the Z axis, instead, while standard deviations re-main constant for E >
256 keV, for smaller energy de-positions the position uncertainties increase with depth.This different behavior can be tracked down to the num-ber of firing segments, since relatively small energy de-positions in the bottom segments are mostly caused bythe Compton scattering of energetic γ rays.In order to understand the factors contributing to theposition resolution and the role of segment geometry, thetotal position fluctuations ∆ are analyzed as a functionof the distance from the edge of the segments. Since theADL bases are simulated assuming the nominal geome-try of the HPGe crystals [1, 8], which may slightly differfrom the real shape of the detectors, the precise positionof the grid points inside the physical crystals is unknown.The distance from the border of the segments is, there-fore, defined within the PSA-grid framework, making theminimum distance equal to the 2-mm grid step. Differ-ent behaviors are expected for interaction points nearthe borders and in the center of the segments, due to thefact that the FoM is calculated considering not only thenet-charge signals but also the transient ones. Figure 6presents a comparison of ∆ distributions obtained for thetwo extreme cases (i.e. segment edge and segment cen-ter). Both distributions can be well reproduced by a sumof Gaussian functions. Two Gaussians with σ = 1 mmand σ = 6 mm are necessary to describe the distribu-tion for the points at the center of the segment, while forthose at the border an additional long-range contributionwith σ = 15 mm is needed. While σ is probably relatedto the size of the grid dimensions – and, consequently, tothe high statistics in the ∆ = 0 bin – σ = 6 mm may berelated to statistical fluctuations. Indeed, the sensitivityof the PSA is expected to be reduced close to the centerof the segments, where the transient signals have the low-est amplitudes. The bootstrapping-induced fluctuationsare probably similar to the signal fluctuations caused byelectronic noise. For the positions close to the segmentsedge, instead, the importance of long-range fluctuationsin the distribution is surprising. In fact, in the vicinity ofthe border, the shape of the pulses – in particular of thetransient-charge ones – rapidly varies and the sensitivityof the PSA is very high. On one hand, such a behaviormay have a geometrical origin and, as discussed in thefollowing, be attributed to the very asymmetric shape ofthe distributions of ∆ X , ∆ Y and ∆ Z . On the other hand,it may be related to the ADL bases themselves. In fact,it has recently been observed that when the distance be-tween a PSA position and the virtual edge of the segmentbecomes lower than 0.5 mm, the shape of the simulatedtransient signal suddenly changes and starts to resemblethat of the firing segment [38]. This clearly requires moreinvestigation, but, if confirmed, could offer an explana- -4 -3 -2 -1
0 10 20 30 40 C oun t s / mm ∆ [mm] Distance = 2 mmDistance = 10 mm σ =1, σ =6 σ =1, σ =6, σ =1502040 0 2 4 6 8 10 ∆ [ mm ] Distance [mm]
FIG. 6: (Color online) Distributions of total position fluctu-ations for the shortest (blue) and the longest (red) distancefrom the edge of a segment. The distribution integrals werenormalized to 1 in order to simplify the comparison. Thecurves represent the functions used to fit the two distribu-tions as described in the text: a sum of 3 Gaussian functionsfor the shortest distance (dashed blue line) and a sum of 2Gaussian functions for the longest distance (dashed red line).In the inset, the fluctuations are presented as a function ofthe distance from the segment edge. -6 -5 -4 -3 -2 -1 C oun t s / mm / mm -40-2002040 ∆ X [ mm ] -40-2002040 ∆ Y [ mm ] -40 -20 0 20 40 Position X [mm] -20-1001020 ∆ Z [ mm ] -40 -20 0 20 40 Position Y [mm]
Position Z [mm]
FIG. 7: (Color online) Position fluctuations as a function ofthe PSA position, identified from the original experimentalsignals. The different patterns are caused by the geometryof the segments and their positions with respect to the frameof reference shown in Figure 2. For each position the maxi-mum intensity was normalized to 1 in order to simplify thecomparison. tion for the large residuals observed at segment borders.However, this result does not represent a failure of thebootstrapping approach. The technique, indeed, allowsto infer the statistical properties of the PSA procedureas it is, pointing out possible systematic issues.In order to study in more detail the effects of theinteraction-point position on the spatial resolution, onecan analyze how the PSA uncertainties evolve inside thesegments of the AGATA detectors. Until now, the posi-tion resolution has been discussed assuming the dimen-sions of the segments as the only geometrical condition.However, since the firing segments are identified by thenet charge, the uncertainties on the interaction-point po-sitions cannot go outside of the segment boundary. Thisis, in fact, what can be observed in Figure 7, whichpresents how the ∆ X , ∆ Y and ∆ Z position fluctuationsdepend on the PSA position of the original experimen-tal signals. The different patterns reflect the geometryof the various segments, and in particular their positionwith respect to the internal frame of reference shown inFigure 2. For example, due to the symmetry of the de-tector and to the fact that the bootstrap position has tobe within the firing-segment volume, the ∆ X fluctuationsmust be negative for positive values of X and vice versa .For the same reason, the evolution of the ∆ Z fluctuationsas a function of the Z coordinate reflects the vertical seg-mentation of the detector, with sign changes occurring atthe segment borders.Therefore, as the fluctuation distributions are position-dependent, the position resolution cannot be treated as FIG. 8: (Color online) Position-uncertainty map for the A ~X direction are estimated for the energy range 64 keV < E p ≤
128 keV. Colors and sizes of theboxes reflect the magnitude of the standard deviation. a general feature anymore, but it rather needs to be de-fined locally. Moreover, since the width of the fluctu-ation distribution rapidly varies with the distance fromthe segment border, asymmetric uncertainties have to beintroduced. For each axis direction k , the lower and up-per uncertainties were defined as the standard deviationof the negative and positive side of the respective dis-tribution of ∆ k , assuming ~r i as the expected value (i.e.∆ k = 0). This resulted in a multi-dimensional map ofAGATA detectors, in which 6 energy-dependent valuesof uncertainty are provided for every grid position. Anexample of this map is shown in Figure 8 for the segment A IV. CONCLUSIONS
The possibility of performing pulse-shape analysis is anessential feature of new-generation HPGe arrays for high-resolution γ -ray spectroscopy studies. A precise identi-fication of the γ -ray interaction point is not only nec-essary for a more accurate Doppler correction, but it isalso crucial when reconstructing the radiation path insidethe detectors using tracking algorithms. In this context,an evaluation of the position uncertainty provides a pos-sibility to improve the tracking algorithms by assigningproper weights to the identified positions.In the present paper, the position resolution of AGATAdetectors was studied via the bootstrapping technique.Despite being demanding in terms of computational re-sources and time, the simplicity of this statistical methodallows to investigate the properties of the position uncer-tainty using practically any data sets (e.g. in-beam data, radioactive sources, background, etc.). Moreover, it per-mits to infer statistical features of a given hypothesis, inthis case the whole PSA procedure, independently fromexternal assumptions.The dependence of the position resolution on variousvariables was analyzed. In particular, the relation be-tween the position uncertainty and the deposited en-ergy, observed previously in a study using the Doppler-correction method, has been confirmed. Since the PSAcompares single-hit traces and experimental overlappingsignals, the position uncertainty increases with the num-ber of firing segments. Surprisingly, for the events whenonly one segment per crystal measures a net signal, thePSA result is identical for the original signals and forthose generated using the bootstrapping technique. Dueto the dimensions of the adopted grid, the FoM presentsa very deep minimum, which is not affected by the fluctu-ations introduced via the bootstrapping method or by aneventual jitter of the signals. While investigating the re-lation between the position fluctuations and the distancefrom the edge of the segment, an unexpected behaviorwas observed for the positions at the segment border.This may be traced back to the pulses of the ADL bases,but it would require a dedicated study. The geometryof the segments and, in particular, the interaction-pointposition were shown to play a crucial role in the posi-tion resolution. As supposed in previous works, not onlythe position fluctuations do not present a Gaussian dis-tribution, but their distributions are position-dependentand they exhibit an important asymmetry for PSA posi-tions near the segment border. This work has been con-cluded by mapping AGATA detectors in terms of energy-dependent asymmetric standard deviations of interactionpositions.Further work is necessary to extend the study of thedeposited-energy dependence to higher energies, in or-der to clearly observe the effects of the pair-productionmechanism in the formation of the experimental signals.Moreover, in order to avoid the systematic bias related tothe number of firing segments and to study the generalproperties of AGATA detectors, the present study shouldbe repeated with a radiation source placed not only infront of the detector – standard condition, similar to thein-beam measurements – but also at its bottom.A natural continuation of this work would be the devel-opment of γ -ray tracking algorithms that use the infor-mation on position resolution to attribute weights to indi-vidual interaction points. Additionally, taking advantageof the information provided by the map of the positionuncertainties, the future PSA basis for AGATA can bedefined on an irregular-geometry grid. Such a solution,already partially adopted for the GRETINA/GRETA de-tectors [39], allows reaching higher position sensitivity incertain detector regions, while limiting the “useless” iter- ations where the position resolution cannot be improved.On the other hand, the demonstrated dependencies ofthe position resolution may provide an impulse to studydifferent segmentation schemes for the next generation ofHPGe detectors. Acknowledgement
The authors would like to thank the AGATA collab-oration and the GANIL technical staff. The excellentperformance of the AGATA detectors is assured by theAGATA Detector Working Group. The AGATA projectis supported in France by the CNRS and the CEA. Thiswork has been supported by the OASIS project no. ANR-17-CE31-0026. A.G. acknowledges the support of theFondazione Cassa di Risparmio Padova e Rovigo underthe project CONPHYT, starting grant in 2017. The au-thors are also grateful to CloudVeneto [40] for the use ofcomputing and storage facilities. [1] S. Akkoyun et al. , Nucl. Instr. and Meth. A , (2012)26.[2] S. Paschalis et al. , Nucl. Instr. and Meth. A , (2013)44.[3] W. Korten et al. , Eur. Phys. J. A , (2020) 137.[4] A. Gadea et al. , Nucl. Instr. and Meth. A , (2011)88.[5] N. Pietralla et al. , Eur. Phys. J.: Web Conf. , (2014)02083.[6] N. Lalovi´c et al. , Nucl. Instr. and Meth. A , (2016)258.[7] E. Cl´ement et al. , Nucl. Instr. and Meth. A , (2017)1.[8] B. Bruyneel et al. , Eur. Phys. J. A , (2016) 700.[9] B. Bruyneel et al. , Nucl. Instr. and Meth. A , (2006)764.[10] B. Bruyneel et al. , Nucl. Instr. and Meth. A , (2009)196.[11] B. Birkenbach et al. ,Nucl. Instr. and Meth. A , (2011)176.[12] B. Bruyneel et al. , Nucl. Instr. and Meth. A , (2011)92.[13] A. Wiens et al. , Eur. Phys. J. A , (2013) 47.[14] F.C.L. Crespi et al. , Nucl. Instr. and Meth. A , (2008)440.[15] M.R. Dimmock et al. , IEEE Transactions on Nuclear Sci-ence , (2009) 1593.[16] T.M.H. Ha et al. , Nucl. Instr. and Meth. A , (2013)123.[17] N. Goel et al. , Nucl. Instr. Meth. A , (2013) 10.[18] M. Ginsz et al. , Proceeding of the 4th InternationalConference on Advancements in Nuclear Instrumenta-tion Measurement Methods and their Applications (AN-IMMA), Lisbon, 2015 , (2015) 1.[19] A. Hernandez-Prieto et al. , Nucl. Instr. and Meth. A ,(2016) 98. [20] T. Habermann et al. , Nucl. Instr. and Meth. A ,(2017) 24.[21] B. de Canditiis et al. , Eur. Phys. J. A , (2020) 276.[22] P. D´esesquelles et al. , Phys. Rev. C , (2000) 024614.[23] P. D´esesquelles, Nucl. Instr. and Meth. A , (2011)324.[24] P. D´esesquelles et al. , Nucl. Instr. and Meth. A ,(2013) 198.[25] H. Li et al. , Eur. Phys. J. A , (2018) 198.[26] F. Recchia, Ph.D. Thesis, Universit`a degli Studi diPadova, 2008.[27] L. Lewandowski et al. , Eur. Phys. J. A , (2019) 81.[28] F. Recchia et al. , Nucl. Instr. and Meth. A , (2009)60.[29] F. Recchia et al. , Nucl. Instr. and Meth. A , (2009)555.[30] P.-A. S¨oderstr¨om et al. , Nucl. Instr. and Meth. A ,(2011) 96.[31] T. Steinbach et al. , Eur. Phys. J. A , (2017) 23.[32] J. Ljungvall et al. , Nucl. Instr. and Meth. A , (2020)163297.[33] A. Lopez-Martens et al. , Nucl. Instr. and Meth. A ,(2004) 454.[34] B. Efron et al. , Statist. Sci. , (1986) 54.[35] H. Varian, Math. J. , (2005) 768.[36] M. Descovich, et al. , Nucl. Instr. and Meth. A ,(2005) 199.[37] R. Venturelli et al. , INFN-LNL Annual Report ,(2004) 220.[38] B. de Canditiis et al. , Contribution at the 19 th AGATAweek and 3 rd Position Sensitive Germanium Detectorsand Application Workshop, Strasbourg, 2018 .[39] A. Korichi et al. , Eur. Phys. J. A (2019) 121.[40] P. Andreetto et al. , Eur. Phys. J.: Web Conf.214